| L(s) = 1 | − 3-s − 2i·5-s + 2i·7-s + 9-s − 4i·11-s + (3 + 2i)13-s + 2i·15-s + 6·17-s + 2i·19-s − 2i·21-s − 4·23-s + 25-s − 27-s − 6·29-s − 2i·31-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.894i·5-s + 0.755i·7-s + 0.333·9-s − 1.20i·11-s + (0.832 + 0.554i)13-s + 0.516i·15-s + 1.45·17-s + 0.458i·19-s − 0.436i·21-s − 0.834·23-s + 0.200·25-s − 0.192·27-s − 1.11·29-s − 0.359i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2496 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.832 + 0.554i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2496 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.832 + 0.554i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.535193162\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.535193162\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 13 | \( 1 + (-3 - 2i)T \) |
| good | 5 | \( 1 + 2iT - 5T^{2} \) |
| 7 | \( 1 - 2iT - 7T^{2} \) |
| 11 | \( 1 + 4iT - 11T^{2} \) |
| 17 | \( 1 - 6T + 17T^{2} \) |
| 19 | \( 1 - 2iT - 19T^{2} \) |
| 23 | \( 1 + 4T + 23T^{2} \) |
| 29 | \( 1 + 6T + 29T^{2} \) |
| 31 | \( 1 + 2iT - 31T^{2} \) |
| 37 | \( 1 - 8iT - 37T^{2} \) |
| 41 | \( 1 + 6iT - 41T^{2} \) |
| 43 | \( 1 - 4T + 43T^{2} \) |
| 47 | \( 1 - 8iT - 47T^{2} \) |
| 53 | \( 1 + 2T + 53T^{2} \) |
| 59 | \( 1 + 8iT - 59T^{2} \) |
| 61 | \( 1 - 14T + 61T^{2} \) |
| 67 | \( 1 - 14iT - 67T^{2} \) |
| 71 | \( 1 + 12iT - 71T^{2} \) |
| 73 | \( 1 + 4iT - 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 - 83T^{2} \) |
| 89 | \( 1 + 6iT - 89T^{2} \) |
| 97 | \( 1 + 12iT - 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.731745668981700008805041037548, −8.290957258064052030807315800673, −7.40264148493251472674620195678, −6.10656792631279968783440983875, −5.84493098539620719939132801873, −5.11133561297271666595882625101, −4.05018025733437630802383425963, −3.23082571355565177455874664371, −1.76868799797709868607325569700, −0.76797061411230491998235014753,
0.923707052934296389807775581667, 2.19511654326418197002835066297, 3.44519621028222968550446406851, 4.06582880167529196429098821110, 5.18005566161846055230581797313, 5.90238378484694340642133581599, 6.80037872016815238477942710644, 7.36827919908318488154581296443, 7.926182954096184027848888407710, 9.127133899257259097977295682522
